Simplify and expand the following expression: $ \dfrac{3q + 4}{2q - 6}-\dfrac{5q - 7}{q + 10} $
Explanation: In order to subtract expressions, they must have a common denominator. Get both fractions over a common denominator of $(2q - 6)(q + 10)$ Multiply the first term by $\dfrac{q + 10}{q + 10}$ $ \begin{align*} \dfrac{3q + 4}{2q - 6} \times \dfrac{q + 10}{q + 10} & = \dfrac{(3q + 4)(q + 10)}{(2q - 6)(q + 10)} \\ & = \dfrac{3q^2 + 34q + 40}{(2q - 6)(q + 10)}\end{align*} $ Multiply the second term by $\dfrac{2q - 6}{2q - 6}$ $ \begin{align*} \dfrac{5q - 7}{q + 10} \times \dfrac{2q - 6}{2q - 6} & = \dfrac{(5q - 7)(2q - 6)}{(q + 10)(2q - 6)} \\ & = \dfrac{10q^2 - 44q + 42}{(q + 10)(2q - 6)}\end{align*} $ Now we have: $ = \dfrac{3q^2 + 34q + 40}{(2q - 6)(q + 10)} - \dfrac{10q^2 - 44q + 42}{(q + 10)(2q - 6)} $ Now both terms have a common denominator we can subtract the numerators: $ = \dfrac{3q^2 + 34q + 40 - (10q^2 - 44q + 42)}{(2q - 6)(q + 10)} $ $ = \dfrac{3q^2 + 34q + 40 - 10q^2 + 44q - 42}{(2q - 6)(q + 10)} $ $ = \dfrac{-7q^2 + 78q - 2}{(2q - 6)(q + 10)}$ Expand the denominator: $ = \dfrac{-7q^2 + 78q - 2}{2q^2 + 14q - 60}$